Cubic anharmonicity

It is expedient first to analyze the contribution from cubic anharmonicity in the simplest case, i.e. for a system free of lateral interactions, when nK and are... 

Figure 13. Plot of potential energy vs. distortion coordinate for a material with a harmonic potential and a material with an additional cubic anharmonic term.

H is the molecular hamiltonian in the absence of the field. This anharmonic energy profile is plotted in Figure 2 for three choices of 2 A/t. A taylor series expansion of this equation around the equilibrium polarization, Vo, gives the effective cubic anharmonicity in the potential, where V replaces the classical position x... 

Interestingly, this peaks for total polarization on the donor or acceptor (V = 1) showing that the cubic anharmonicity is related to the ground state polarization in this two-orbital case. When the square of the ground state polarization is not too large, the anharmonicity is simply proportional to the polarization of the ground state so from eqs 9b and 17 in this simple example... 

To explain the idea of the method [13,14], let us consider a two-phonon decay of a highly excited local mode caused by the interaction /7irl( = QY.rf V3 A,, where V3 lI/ are the cubic anharmonicity interaction parameters, Q is the coordinate... 

EAS (Engler, Andose, Schleyer) [184] is quite an old force field designed to model alkanes exclusively. The harmonic potential is used for the bond stretching and cubic anharmonic for the valence angle bending. No out of plane, electrostatic or cross terms are included. The nonbonded interactions are represented by the Buchingham potential. 

EFF (Empirical force field) [186] has been designed just for modeling hydrocarbons. It uses the quartic anharmonic potential for the bond stretching, and the cubic anharmonic for the valence angle bending. No out of plane or electrostatic terms are involved, although the cross terms, except torsion-torsion and bend-torsion ones, are included. 

MM2 [189] uses cubic anharmonic potential to represent the bond stretching, up to sixth power expansion for the valence angle bending, and harmonic field for the out-of-plane deformations. The stretch-bending cross term is included. 

The two terms in the formula (24) for x have a simple physical interpretation the first is a correction to the energy levels owing to quartic anharmonicity in the first order of perturbation, and the second is a correction due to cubic anharmonicity in the second order of perturbation, It is well known that these... 

The second term in aB arises from second-order perturbation theory it may be described as due to the fact that cubic anharmonicity results in a mean first-power displacement in q given by... 

It is clear from equation (61) that if the harmonic force field of a molecule is known, each observed af constant gives a linear relation between the cubic anharmonic constants m, where s ranges over all normal modes in the molecule (subject to symmetry restrictions). 

The observed a values are of course generally an important source of information on the cubic anharmonic force field. However, in the presence of a Coriolis resonance the particular a values involved are dominated by the harmonic Coriolis contribution arising from equation (66), and analysis of a Coriolis resonance essentially gives information on the constant or... 

Often the original structure determination will have involved some uncorrected vibrational averaging effects it may be an r0 or an r, structure.28 However, once /2, or some approximation to /3, has been determined it is possible to correct r to re and obtain an improved equilibrium structure (in most cases this correction can be made directly from the other isotopic species, etc.). Similarly, it is often true that the harmonic field /2 is calculated from the observed fundamentals (the v values) rather than the harmonic vibration wavenumbers (the to values), for want of information on the corrections. However, once /4, or some approximation to/4, has been determined, it may be used to calculate a complete set of x values and hence to calculate all the corrections to obtain the co values. Thus the calculation of re and may be improved from a knowledge of /3 and /4. 

The anharmonic response can be obtained by inclusion of the cubic anharmonicity in the vibrational potential,... 

Figure 3 Calculated vibrational cascade in crystalline naphthalene at T = 0, for initial excitation at 1627 cm-1. The calculation uses Equation (6), which assumes that cubic anharmonic coupling dominates. From Ref. 5. 

Figure 4 Average energy of the nonequilibrium vibrational population distribution computed for the vibrational cascade in crystalline naphthalene in Fig. 3. At T = 0, the peak moves toward lower energy at a roughly constant rate, the vibrational velocity of 8.9 cm-1 ps. The initial 1627 cm-1 of vibrational energy is dissipated in 180 ps. The vibrational velocity is the same at 300 K. In the limit that cubic anharmonic coupling dominates [Equation (6)], increasing the temperature increases the rates of up- and down-conversion processes, but has no effect on the net downward motion of the population distribution. Although the lifetimes of individual vibrational levels will decrease with increasing temperature, VC is not very dependent on temperature in this limit. (From Ref. 5.)...

Since the combination band anharmonicity (<10 cm 1) (84) is less than our spectral resolution ( 60 cm-1 in this measurement), excitation of the combination C-C stretch and C=N stretch is seen as excitation of both fundamentals (44). Pumping the combination band causes an instantaneous jump in the populations of the C-H stretch, C-C stretch, and C=N stretch. The directly pumped C=N stretch excitation rises to a level about 20 times greater than when it is indirectly populated with C-H stretch pumping. A C=N stretch and a C-C stretch on the same molecule can annihilate each other via cubic anharmonic coupling to create C-H stretch excitations (47). 

We turn now to calculation of the lifetimes of the vibrational modes. We consider only the contribution of cubic anharmonic terms in the potential energy to the lifetime, an assumption that is vahd at sufficiently low temperatures. The energy transfer rate from mode a, Wa, can be calculated with the Golden Rule, written as the sum of terms that can be described as decay and collision [ 143], the former typically very much larger except at low frequency where both terms are comparable. The anharmonic decay rate of vibrational mode a is then the sum of these two terms ... 

Dlott and Payer s doorway model of phonon pumping [50] suggested two essentially equivalent methods for determining the rate of phonon pumping of doorway vibrations for large polyatomic molecules where only two phonons are needed to pump a doorway vibration. The anharmonic term in the Hamiltonian responsible for two-phonon doorway mode pumping is a cubic anharmonic term of the form [21,50],... 

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